A NEW IMPROVED QUANTUM EVOLUTIONARY ALGORITHM WITH MULTIPLICATIVE UPDATE FUNCTION

International Journal on Computational Science & Applications (IJCSA) Vol.6,No.1,February 2016
DOI:10.5121/ijcsa.2016.6102 17
A NEW IMPROVED QUANTUM EVOLUTIONARY
ALGORITHM WITH MULTIPLICATIVE UPDATE
FUNCTION
Mohsen Zamania, Mehdi Alinaghiana and M.Ali Yazdania
Department of Industrial and Systems Engineering, Isfahan University of Technology,
84156-83111 Isfahan, Iran
ABSTRACT
The Quantum Evolutionary Algorithm (QEA) is a new subcategory of evolutionary computation in which
the principles and concepts of quantum computation are used, and to display the solutions it utilizes a
probabilistic structure.Therefore, it causes an increase in the solution space. This algorithm has two major
problems: hitchhiking phenomenon and slow convergence speed.In this paper, to solve the problems a
multiplicative update function called quantum gate is proposed that in addition to considering the best
global solution ، considers the best solution of each generation. The results of one max and knapsack
problems and five famous numerical functions show that the proposed method has a significant advantage
compared with the basic algorithm in terms of performance, quality of solutions and convergence speed.
KEYWORD
quantum evolution algorithm, multiplicative update function,one maxproblem, knapsack problem.
1.INTRODUCTION
Evolutionary algorithms (EAs) are principally aStochastic Search and optimization method based
on theprinciples of Darwin's theory of evolution[1]. Compared with traditional optimization
methods, such as calculus-based methods and gradient-based methods EAs are robust.Most of the
methods have the problemof trapping in local optimum whereasEAs solve this problem to some
extent[2].In EA sthe parameters must be set so that the balancing between movingtowards the
best solution and searching for better solutions is preserved. Focus on the best answer increases
the Possibility of trapping in local optimumwhile focus on the searching for better solutions
increases thesolution time.
Quantum computing was first introduced by Benioff and Feynman in early 1980s[3]. In recent
years،many efforts on the theory and design of algorithms and quantum computers have
progressed.The research iscarried out in order to achieve higher computing power compared with
conventional computers.As well asQuantum computing is employed in producing circuits of
quantum computers[4].The concepts of quantum computing was successfully used to improve the
efficiency of evolutionary algorithms on traditional PCs and led to the development of quantum
evolutionary algorithms (QEA).Han and Kim proposed the QEA to solve the combinatorial

18
optimization problems[5]. QEA uses quantum computing concepts such as quantum bit, linear
superposition of states and quantum rotation gate and is the first approach that enables the use of
quantum representation instead of conventional binary, numeric or symbolic representations.
Platel and colleagues have investigated the weaknesses of QEA and detected The problem of
premature convergence, and to solve this problem that caused by the Hitchhiking Phenomenon
they proposed versatile quantum-inspired evolutionary algorithm(VQEA)[6]. Hitchhiking is a
Phenomenon in which the amount of one of the bits generated by the optimization algorithm is
incompatible with the expected value but by the amount of general fitness of global solution، the
value of this bit remains unchanged during the run of the algorithm.
An advantage of the VQEA is that the Information obtained about the search space during the
evolution of population does not remain at the level of one individual but shared Between
members of the population.
Many papers have used QEA in solving combinatorial optimization problems[7-9].
In this paper the Quantum Evolutionary Algorithm with multiplicative Update function is
presented where the value of Q-bits updated by a multiplicative function. The inputs of function
are: present solution, the global best solution and the best solution to each generation and the
output is rotation angle.This paper is organized as follows.Section 2 describes the Quantum
evolutionary algorithms. Section 3 explains the proposed algorithm. Section 4 summarizes the
results. Finally, concluding remarks follow in Section 5.
2.QUANTUM EVOLUTIONARY ALGORITHM
2.1.Concepts
QEA uses quantum computing concepts such as quantum bit, linear superposition of states and
quantum rotation gate and is the first approach that enables the use of quantum representation
instead of conventional binary, numeric or symbolic representations. Concepts of QEA are
explained in the following.
• Quantum bit: Quantum bit or Q-bit is defined as the smallest unit of data storage which
can be in the state of "0", in the state of "1" or a superposition of two states. Each Q-bit is
defined by a pair of values ቂ
ߙ
ߚቃ The α2
is the probability of state 0 and β2
is the probability
of state 1. Equation (1) expresses the relationship between these two values
(1)2 2
1α β+ =
• Q-bit individual: Q-bit individual is defined as a string of Q-bits. Each Q-bit individual
can represent a linear superposition of states, probabilistically. In other words, each

19
individual Q-bit with m bits can represent 2௠
state simultaneously.For example,consider
the following Q-bit individual with 3 Q-bits.
The Q-bit individual is able to represent the eight states “000”،”001” ،”010” ،”011” ،”100”
،”101” ،”110” ،”111” with the possibilities of 1/12،1/6،1/12،1/6،1/12،1/6،1/12،1/6
Respectively.
Each Q-bit individual considered as a distribution of all possible solutions in the search space.
The Q-bit representation has a better characteristic of generating diversity in population than any
other representations like Binary.
• Quantum gate: In a QEA, Q-gate updates the values of Q-bits. Using Q-gate operator on
each Q-bit simulates the evolutionary processes in the population.After each update by
Q-gate, Q-bit values must satisfy relationship (1). To update the values of Q-bits,
quantum rotation gate operator must be used in accordance with relationships (3) and (4).
)3(( )
cos sin
U
sin cos
θ θ
θ
θ θ
∆ − ∆ 
∆ =  ∆ ∆ 
In the above relationship, ∆θ is the Q-bit angle of rotation to state 0 or state 1. At each iteration,
Q-gate operator updates Q-bits in accordance with equation (4).
(4)
In the above relationship, ,i jα and ,i jβ are the parameters related to the j-th dimension of i-th
Q-bit individual.
( )
( )
( )
( )
, ,
, ,
t+1 t
( )*
t+1 t
i j i j
i j i j
U
α α
θ
β β
   
=   
   

20
2.2.The Structure of Algorithm
According to the definitions above،QEA pseudo-code is as follows:
Procedure QEA
begin
t=0
initialize Q(t)
make P(t) by observing Q(t)
evaluate P(t)
store the best solutions among P(t) into b
while (not termination-condition) do
begin
t = t+1
update Q(t) using Q-gate
make P(t) by observing the states ofQ(t-1)
evaluate P(t)
store the best solutions among P(t) into b
end
end
In the step of “initialize Q(t) ,” ( ), ti jα and ( ), ti jβ of all ( ), ti jq are initialized with 1
2
.It
means that one Q-bit individual, represents the linear superposition of all the possible states with
the same probability.in the algorithm، n is the string length of the Q-bit individual, and is
determined according to the dimensions of solution, and m is the number of Q-bit individuals that
indicates the population.
The next step makes binary solutions in P(0) by observing the states of Q(0), where binary
solution of each Q-bit individual is created at generation t=0 according to the ( ), ti jα and ( ), ti jβ
.onebinary solution ,Pi
0
، i=1,2,…,n , is a binarystring of length m, which is formed by
selectingeither 0 or 1 for each bit using the probability of qi(0). in each stepto make a solution
،equation (5) is used as follow:
(5)
( ) 2
1 0,1
0
ij
ij
if rand
p
otherwise
α ≥
= 

In this step Each binary solution Xi
0
is evaluated to give a measure of its fitness and the initial
best solutions are then selected among the binary solutions P(0), and stored into until the
termination condition is satisfied, QEA is running in the while loop as follows:

21
• In the while loop, binary solutions in p(t) are formed by observing the states of Q(t-1)
According to equation (6).
• 2.After generation of new solutions،each binary solution is evaluated for the fitness
value.
• 3.Q-bit individuals in Q(t) are updated by Q-gates so that the updated Q-bit should satisfy
the equation(1). equation(2),(3) is used as a basic Q-gate in QEA where ∆ߠ is the rotation
angle of each Q-bit that can be obtained from Table (1)[10].
Table 1: Rotation angle in QEA algorithm
0false00
0true00
false10
0true10
false01
0true01
0false11
0true11
Parameters presented in Table (1) are specifically for a minimization problem. In this table, ip
and ib are the i-th dimension of current answer and best answer respectively; ∆θis the rotation
angle and s denotes the angle. In the basic quantum algorithm, α and β can take any quantity
within the unit circle 4.The best solution is stored in b.
2.3.The Proposed Algorithm
The classical QEA described in the previous section.The result of researches on QEA shows that
this algorithm has the premature convergence problem caused by Hitchhiking phenomenon. This
phenomenon occurs when the amount of one of the bits generated by the optimization algorithm
is incompatible with the expected value But by the amount of general fitness of global solution،
the value of this bit remains unchanged during the implementation(execution) of the algorithm, it
( , )t t
ij ijs α βt
ijθ∆( ) ( )F p F b≥ibip
, 0t t
ij ijα β <, 0t t
ij ijα β ≥
1±1±
1±1±
1−1+0.01π
1±1±
1+1−0.01π
1±1±
1±1±
1±1±

22
means that The bit of interest(concerned bit)، take a hitchhiker from other bits of global solution
to solve this problem and improve the QEA، some solutions have been proposed, and in the
following the description of each one is provided.
• Generate multiple solution by observing every individual
some changes have been made For greater diversity of solutions Such that each Q-bit individual
‫ݍ‬௜ሺtሻ ∈ Qሺtሻ generates L solution.Then the best solution is stored in the Present solution ܿ௜ሺtሻ. in
each generation the best individual (solution) is selected and stored as the current best solution in
zሺtሻ, also the best global solution of b in each generation is updated.
• The use of current best solution to each generation
In this paper, to solve the problem of Hitchhiking phenomenon in the Q-bits update step، In
addition to the global best solution، current best solution in each iteration is considered.To update
Q(t) ، each Q-bit individual is updated According to (3) , (4) .The rotation angle is calculated by
the formula provided below.
The simplified equation is achieved as
In equation (7)،ߛଵand ߛଶ are two positive Coefficient that shows Confidence measure in global
best solution and best solution of each generation respectively.To speed up the convergence,the
algorithm designed in such a way that the bits that are similar to optimal global and current
solutions bits، converge to their value faster than other bits. To consider this issue, the coefficient
α that is a positive factor and greater than 1 is determined.equation (7) is defined so that The
positive values of ∆θ increase the probability of 1 and negative values increase the probability of
0.
It should be mentioned that in proposed Algorithm ߙ andߚ take only the positive
values.According to Figure (1)،the α , β range is only in the first quarter circle trigonometryThe
technique presented in [4] has been used to avoid premature convergence, so the Q-bits cannot be
closer to 0 or 1 than a specified ε value.

23
Figure1:range of rotation Angle
The proposed algorithm pseudo-code is given below:
procedure QEA
t ←0
Initialize population Q(t)
while (not termination condition) do
t ←t +1
make P(t) by observing Q(t)
for each Q-bit individual qi (t) do
Run l times by observingqi(t)
ci(t)← best solution among l of times
end for
evaluate P(t)
z (t)← best solution among P(t)
Update b
Update Q(t) using Q-gate by P(t), z (t), b, ci(t)
end while
end procedure

24
3.EXPERIMENTS AND NUMERICAL RESULT
To demonstrate the performance and behavior of the proposed algorithm،one max and knapsack
Problems and five other Numerical Functions are utilized.
3.1.Parameter Adjustment
Taguchi is a Powerful statistical method that is used to set parameters. In this method Factors are
divided into two categories: controllable factors and Stochastic factors. Stochastic factors have
inevitable effects.so Efforts are made tominimize the impact of these factors. controllable factors
are determined at such levels that the method efficiency is at its maximum and its stability is
preserved.In taguchi،instead of the value of solution، The ratio of the signal to noise S / N is used
to examine the solution. In This ratio S is the utility (objective function value) and N is
Undesirability (The standard deviation of the objective function values)[11].As a result، the aim
is to increase the value of this ratio as possible.The proposed algorithm has 3 parameters that
must be known before implementation.Each of the parameters are investigated at three
levels.Table (2) shows the parameters and levels of assessment.
Table2: parameters and levels of assessment
parameterlevelvalue
1γ
10.2*ߨ
20.25*ߨ
30.3*ߨ
2γ
10.1*ߨ
20.15*ߨ
30.2*ߨ
α
11
21.3
31.5
Data analysis was performed by Minitab software. According to the number of selective factors
and levels for the analysis، Standard orthogonal table L9(3*3)was selected.Five problems of one

25
max and Five problems of knapsack were randomly chosen in order to investigate each of the
vectors.each problem accordingto the intended values for parameters in each row of the Taguchi
table that related to the Essence of the problem, was performed three times. the amount of error
was considered as the criterion of solution to solve the problems.Then based on Taguchi
performance in Minitab, in each problem for each parameter, the level that had the highest S / N
ratio was selected as the best level for that parameter.Table (3) shows the value of levels selected
for parameters.
Table3: the final parameters of the problem
N is the dimension of solution
3.2.One Max Problem
In this problem a bit string of length k is Defined.The goal is to maximize the number of ones in
bit string and the global Optimum value is k at x=111…1. To study the behavior of algorithm the
method introduced in [4] is used. Figure (2) shows the Q-bits evolution of the individuals in the
proposed algorithm. In the figure، the Color tones display the Square average value of β (ߚ̅ଶ
).
The black color Refers to ሺߚ̅ሻ ଶ
= 0 and White Refers to ሺߚ̅ሻ ଶ
= 1. With the Departure of Q-bits
to the global optimum، the Q-bits color Varies from gray to white.

Figure (3) shows the best solution of each iteration (generation). As seen in Figure 3
proposed algorithm can find the optimal solution faster than QEA for k = 100. On the other hand,
in running step، sometimes the QEA converges to non
not exist in the proposed algorithm
Figure3:Diagram of the best solution
vertical axis is the best solution
Fitness
Journal on Computational Science & Applications (IJCSA) Vol.6,No.1,February 2016
Figure2: the evolution of Q-bits
shows the best solution of each iteration (generation). As seen in Figure 3
sometimes the QEA converges to non-Optimum solutions. But this problem does
exist in the proposed algorithm
Diagram of the best solution to each iteration (The horizontal axis is iteration number and The
vertical axis is the best solution to each generation)
QEA
IQEA
Fitness
26
shows the best solution of each iteration (generation). As seen in Figure 3، The
Optimum solutions. But this problem does
The horizontal axis is iteration number and The

27
Table (4)shows the numerical results of experiments on one max problem in proposed and classic
algorithm. Each experiment has been performed 10 times, the best and worst solution, and the
average of solutions are given in table below
Table4: the numerical results of one max
QEA IQEA
n best solution solution gap% time solution gap time
100 100
best 100
0 2.13
best 100
0 1.9mean 100 mean 100
worst 100 worst 100
250 250
best 250
0 7.1
best 250
0 6.8mean 249.5 mean 250
worst 249 worst 250
350 350
best 350
0 16.2
best 350
0 15.14mean 348.12 mean 350
worst 344 worst 350
500 500
best 496
0.8 39.9
best 500
0 38.2mean 494 mean 498
worst 491 worst 496
650 650
best 639
1.6 52.3
best 650
0 49.7mean 630 mean 650
worst 622 worst 650
3.3.Knapsack Problem
Knapsack problem which is a well-known combinatorial optimization problem is included in a
class of NP-hard problems [12]. This problem is used in some papers for assessment. In this
problem some objects are given by specified weight and value. The most famous kind of this
problem is the 0–1 knapsack problem.Ifxi=1 , the ith item is selected for the knapsack. The
knapsack problem can be described as selecting asubset of items from among various items so
that it is most profitable, given that the knapsack has limited capacity. Since there are n objects so
2n
possible combinations of objects can be built. The 0–1 knapsack problem is described as
follows:
)8(
1
max( )
n
i i
i
v x
=
∑

28
)9(
Subject to :
1
n
i i
i
w x W
=
≤∑
Consider the knapsack problem with 20 objects. The weight vector and its value is as follows:
It is obvious because of ascending of weight vector and descending of the value vector,the
optimal solution of objects choice, starts from object 1 and then object 2, and continue until the
maximum weight limit allows. it is assumed to W=55. As a result, the optimal solution of choice
of 10 first objects is f=20+19+…+11=155. After the implementation of the proposed algorithm
the evolution of Q-bits
is shown in Figure (4).
Figure4: the evolution of Q-bits
As seen in figure (4), Q-bits find their optimum value quickly. 10 first objects that their total
weight is 55 have the highest worth and goes toward 1 at early generations. The diagram of global
best solution versus iteration (generation) is shown in Figure (5).

Figure5:Global best solution (the horizontal axis represents the number of generations
As can be seen the proposed algorithm converges
The algorithm has been implemented on several
matrices are defined as following:
The experiment conducted on various N and results can be observed in Table (5).Each
experiment is performed 10 times. The best and worth solution and the average of so
given in Table (5).
Fitness
the horizontal axis represents the number of generations and the vertical axis
represents the global best answer)
As can be seen the proposed algorithm converges to the optimum solution with a higher speed.
The algorithm has been implemented on several knapsack problems,weight، worth and capacity
matrices are defined as following:
experiment is performed 10 times. The best and worth solution and the average of so
QEA
IQEA
29
the vertical axis
a higher speed.
worth and capacity
experiment is performed 10 times. The best and worth solution and the average of solutions are

30
Table 5:.results of Knapsack problem
knapsack problem QEA IQEA
N
best
solution
solution gap% time(s) solution gap time
40 240
best 240
0 1.84
best 240
0 0.96mean 236 mean 240
worst 230 worst 240
80 493
best 484
1.8 2.8
best 493
0 1.7mean 465 mean 489
worst 444 worst 474
100 602
best 589
2.1 2.5
best 597
0.8 2.1mean 574 mean 592
worst 544 worst 589
150 932
best 891
4.4 25
best 921
1.1 17.2mean 860 mean 350
worst 831 worst 863
250 1522
best 1478
2.9 24.3
best 1497
1.6 22.8mean 1450 mean 1773
worst 1410 worst 1458
Five Numerical functions such as Sphere, Ackley, Griewank, Rastrigin, Schwefel, Rosenbrock
are applied to evaluate the proposed algorithm. Information on this functions can be seen in Table
(6) and Figure (6).
Table 6: Numerical functions
Minimu
m value
DomainFunction
Function
name
0
[-
100,100
]
2
1
N
i
i
x
=
∑
Sphere
(N=30)
1
0[-32,32]2
1 1
1 1
20exp( 0.2 ) exp( cos(2 ) 20
N N
i i
i i
x x e
N N
π
= =
− − − + +∑ ∑
Ackley
(N=30)
2

31
0
[-
600,600
]
2
1 1
1
cos( ) 1
4000
NN
i
i
i i
x
x
i= =
− +∑ ∏
Griewank
(N=30)
3
0[-5,5]
2
1
10 ( 10cos(2 ))
N
i i
i
N x xπ
=
+ −∑
Rastrigin
(N=30)
4
0[-30,30]
1
2 2 2
1
1
(100( ) ( 1) )
N
i i i
i
x x x
−
+
=
− + −∑
Rosenbroc
k (N=30)
5
Figure6: Numerical functions diagram

32
The results of numerical functions can be observed in Table (7). Each experiment is performed 10
time, the best and worth solution and the average of solutions are given.
Table 7: The results of numerical functions
numerical function QEA IQEA
Function best solution solution solution
Sphere 0
best 35612 best 0
mean 37254 mean 16
worst 44081 worst 44
Akley 0
best 0.2 best 0
mean 1.8 mean 0.04
worst 4.02 worst 0.1
Rastrigin 0
best 32 best 0
mean 43 mean 0
worst 67 worst 0
Griewank 0
best 8.01 best 0
mean 8.6 mean 1.2
worst 9.28 worst 1.3
Rosenbrock 0
best M best 11
mean M mean 18.5
worst M worst 29
During the experiment the run time(duration of solution) of two algorithms was almost equal، But
as shown in Table (7) The proposed algorithm is converged into better solutions. In Table (7), M
is a big number (M>10000).
4.CONCLUSION
In this paper the Quantum Evolutionary Algorithm with multiplicative Update function is
presented.In this function to update the Q-bits in each generation، the best solution and the global
best solution is used.The Q-bit rotation is limited only to the first quarter of trigonometry.it
causes the simplicity of the algorithm and the computational complexity is reduced.Experiments
on onemax and knapsack problems and five numerical functions prove that this algorithm has

33
high convergence speedand because of using two types of solutions، the Hitchhiking
Phenomenon reduces in comparison with the classical algorithm.
5. REFERENCE
[1] Haupt, R. L., & Haupt, S. E. (2004). Practical genetic algorithms. John Wiley & Sons.
[2] Bäck, T., Fogel, D. B., & Michalewicz, Z. (Eds.). (2000). Evolutionary computation 1: Basic
algorithms and operators (Vol. 1). CRC Press.
[3] Kaye, P., Laflamme, R., & Mosca, M. (2007). An introduction to quantum computing.
[4] Han, K. H., & Kim, J. H. (2004). Quantum-inspired evolutionary algorithms with a new termination
criterion, H &epsi; gate, and two-phase scheme. Evolutionary Computation, IEEE Transactions on,
8(2), 156-169.
[5] Narayanan, A., & Moore, M. (1996, May). Quantum-inspired genetic algorithms. In Evolutionary
Computation, 1996., Proceedings of IEEE International Conference on (pp. 61-66). IEEE.
[6] Platel, M. D., Schliebs, S., & Kasabov, N. (2007, September). A versatile quantum-inspired
evolutionary algorithm. In Evolutionary Computation, 2007.CEC 2007. IEEE Congress on (pp. 423-
430). IEEE.
[7] Mahdabi, P., Jalili, S., & Abadi, M. (2008, July). A multi-start quantum-inspired evolutionary
algorithm for solving combinatorial optimization problems. In Proceedings of the 10th annual
conference on Genetic and evolutionary computation (pp. 613-614). ACM.
[8] Zhang, R., & Gao, H. (2007, August). Improved quantum evolutionary algorithm for combinatorial
optimization problem. In Machine Learning and Cybernetics, 2007 International Conference on (Vol.
6, pp. 3501-3505). IEEE.
[9] da Cruz, A. V. A., Vellasco, M. M., & Pacheco, M. A. C. (2008). Quantum-inspired evolutionary
algorithm for numerical optimization. In Quantum inspired intelligent systems (pp. 115-132).
Springer Berlin Heidelberg.
[10] Zhang, G. (2011). Quantum-inspired evolutionary algorithms: a survey and empirical study. Journal
of Heuristics, 17(3), 303-351.
[11] C Montgomery, D. (1997). Montgomery Design and Analysis of Experiments.
[12] Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: a guide to the theory of NP-
completeness. 1979. San Francisco, LA: Freeman.

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